Hardy inequalities for large fermionic systems

Given $0<s<\frac{d}{2}$ with $s\le 1$, we are interested in the large $N$-behavior of the optimal constant ${\kappa }_N$ in the Hardy inequality ${\sum }_{n=1}^N(-{\mathrm{\Delta }}_n{)}^s\ge {\kappa }_N{\sum }_{n<m}\mathrm{\mid }{X}_n-X_m{\mathrm{\mid }}^{-2s}$, when restricted to antisymmetric functions. We show that $N^{1-\frac{2s}{d}}{\kappa }_N$ has a positive, finite limit given by a certain variational problem, thereby generalizing a result of Lieb and Yau related to the Chandrasekhar theory of gravitational collapse.